Theorem signslema 28392

Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)

Hypotheses

Ref	Expression
signslema.1	|- ( ph -> E e. NN0 )
signslema.2	|- ( ph -> F e. NN0 )
signslema.3	|- ( ph -> G e. NN0 )
signslema.4	|- ( ph -> H e. NN0 )
signslema.5	|- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
signslema.6	|- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )

Assertion

Ref	Expression
signslema	|- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Proof of Theorem signslema

Step	Hyp	Ref	Expression
1		signslema.5	. . . . . 6 |- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
2	1	simpld 459	. . . . 5 |- ( ph -> E < G )
3	2	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> E < G )
4		signslema.4	. . . . . . . . . 10 |- ( ph -> H e. NN0 )
5	4	nn0cnd 10860	. . . . . . . . 9 |- ( ph -> H e. CC )
6		signslema.2	. . . . . . . . . 10 |- ( ph -> F e. NN0 )
7	6	nn0cnd 10860	. . . . . . . . 9 |- ( ph -> F e. CC )
8	5, 7	subcld 9936	. . . . . . . 8 |- ( ph -> ( H - F ) e. CC )
9		signslema.3	. . . . . . . . . 10 |- ( ph -> G e. NN0 )
10	9	nn0cnd 10860	. . . . . . . . 9 |- ( ph -> G e. CC )
11		signslema.1	. . . . . . . . . 10 |- ( ph -> E e. NN0 )
12	11	nn0cnd 10860	. . . . . . . . 9 |- ( ph -> E e. CC )
13	10, 12	subcld 9936	. . . . . . . 8 |- ( ph -> ( G - E ) e. CC )
14	8, 13	subeq0ad 9946	. . . . . . 7 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 <-> ( H - F ) = ( G - E ) ) )
15	14	biimpa 484	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( H - F ) = ( G - E ) )
16	15	breq2d 4449	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 0 < ( H - F ) <-> 0 < ( G - E ) ) )
17	6	nn0red 10859	. . . . . . 7 |- ( ph -> F e. RR )
18	4	nn0red 10859	. . . . . . 7 |- ( ph -> H e. RR )
19	17, 18	posdifd 10145	. . . . . 6 |- ( ph -> ( F < H <-> 0 < ( H - F ) ) )
20	19	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( F < H <-> 0 < ( H - F ) ) )
21	11	nn0red 10859	. . . . . . 7 |- ( ph -> E e. RR )
22	9	nn0red 10859	. . . . . . 7 |- ( ph -> G e. RR )
23	21, 22	posdifd 10145	. . . . . 6 |- ( ph -> ( E < G <-> 0 < ( G - E ) ) )
24	23	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> 0 < ( G - E ) ) )
25	16, 20, 24	3bitr4rd 286	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> F < H ) )
26	3, 25	mpbid 210	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> F < H )
27		0red 9600	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 e. RR )
28	22, 21	resubcld 9993	. . . . . 6 |- ( ph -> ( G - E ) e. RR )
29	28	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) e. RR )
30	18, 17	resubcld 9993	. . . . . 6 |- ( ph -> ( H - F ) e. RR )
31	30	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( H - F ) e. RR )
32	2	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> E < G )
33	23	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( E < G <-> 0 < ( G - E ) ) )
34	32, 33	mpbid 210	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( G - E ) )
35		2pos 10633	. . . . . . 7 |- 0 < 2
36		breq2 4441	. . . . . . 7 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> ( 0 < ( ( H - F ) - ( G - E ) ) <-> 0 < 2 ) )
37	35, 36	mpbiri 233	. . . . . 6 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> 0 < ( ( H - F ) - ( G - E ) ) )
38	28, 30	posdifd 10145	. . . . . . 7 |- ( ph -> ( ( G - E ) < ( H - F ) <-> 0 < ( ( H - F ) - ( G - E ) ) ) )
39	38	biimpar 485	. . . . . 6 |- ( ( ph /\ 0 < ( ( H - F ) - ( G - E ) ) ) -> ( G - E ) < ( H - F ) )
40	37, 39	sylan2 474	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) < ( H - F ) )
41	27, 29, 31, 34, 40	lttrd 9746	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( H - F ) )
42	19	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( F < H <-> 0 < ( H - F ) ) )
43	41, 42	mpbird 232	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> F < H )
44	5, 10, 7, 12	sub4d 9985	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) = ( ( H - F ) - ( G - E ) ) )
45		signslema.6	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )
46	44, 45	eqeltrrd 2532	. . . 4 |- ( ph -> ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } )
47		ovex 6309	. . . . 5 |- ( ( H - F ) - ( G - E ) ) e. _V
48	47	elpr 4032	. . . 4 |- ( ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } <-> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
49	46, 48	sylib 196	. . 3 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
50	26, 43, 49	mpjaodan 786	. 2 |- ( ph -> F < H )
51	1	simprd 463	. . . . 5 |- ( ph -> -. 2 || ( G - E ) )
52	51	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( G - E ) )
53	15	breq2d 4449	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 2 || ( H - F ) <-> 2 || ( G - E ) ) )
54	52, 53	mtbird 301	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( H - F ) )
55		2z 10902	. . . . . . 7 |- 2 e. ZZ
56	9	nn0zd 10972	. . . . . . . 8 |- ( ph -> G e. ZZ )
57	11	nn0zd 10972	. . . . . . . 8 |- ( ph -> E e. ZZ )
58	56, 57	zsubcld 10979	. . . . . . 7 |- ( ph -> ( G - E ) e. ZZ )
59		dvdsaddr 13902	. . . . . . 7 |- ( ( 2 e. ZZ /\ ( G - E ) e. ZZ ) -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
60	55, 58, 59	sylancr 663	. . . . . 6 |- ( ph -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
61	51, 60	mtbid 300	. . . . 5 |- ( ph -> -. 2 || ( ( G - E ) + 2 ) )
62	61	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( ( G - E ) + 2 ) )
63		2cnd 10614	. . . . . . 7 |- ( ph -> 2 e. CC )
64	8, 13, 63	subaddd 9954	. . . . . 6 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 2 <-> ( ( G - E ) + 2 ) = ( H - F ) ) )
65	64	biimpa 484	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( ( G - E ) + 2 ) = ( H - F ) )
66	65	breq2d 4449	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( 2 || ( ( G - E ) + 2 ) <-> 2 || ( H - F ) ) )
67	62, 66	mtbid 300	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( H - F ) )
68	54, 67, 49	mpjaodan 786	. 2 |- ( ph -> -. 2 || ( H - F ) )
69	50, 68	jca 532	1 |- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1383   e. wcel 1804   {cpr 4016   class class class wbr 4437  (class class class)co 6281   RRcr 9494   0cc0 9495   + caddc 9498   < clt 9631   - cmin 9810   2c2 10591   NN0cn0 10801   ZZcz 10870   || cdvds 13863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-dvds 13864

This theorem is referenced by: (None)